Problem: $f(x) = \sqrt{ 7 - \lvert x \rvert }$ What is the domain of the real-valued function $f(x)$ ?
$f(x)$ is undefined when the radicand (the expression under the radical) is less than zero. So we know that $7 - \lvert x \rvert \geq 0$ So $\lvert x \rvert \leq 7$ This means $x \leq 7$ and $x \geq -7$ ; or, equivalently, $-7 \leq x \leq 7$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid -7\leq x \leq7\, \}$.